I've Definitely Seen it Before, Part 2

Geometry Level 3

Consider an i s o s c e l e s \color{#E81990}{isosceles} Δ A B C \color{#69047E}{\Delta ABC} , with ∠ A = 2 0 ∘ \color{#20A900}{\angle A=20^{\circ}} , and the other two angles being equal. D D and E E are points on A C ‾ \overline{AC} and A B ‾ \overline{AB} respectively, such that ∠ D B C = 5 0 ∘ \color{#EC7300}{\angle DBC=50^{\circ}} and ∠ E C B = 6 0 ∘ \color{#3D99F6}{\angle ECB=60^{\circ}} . Determine the magnitude of ∠ D E C \angle DEC in degrees.

Note:- This question is not mine in any way whatsoever. It is a quite popular problem, and I could not find the exact source, and will be eagerly waiting for the same. Protractor users will be prosecuted.

18 27 30 15

Satvik Golechha by Satvik Golechha , 21, India

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1 solution

Mathh Mathh
Aug 9, 2014

This problem is called Langley's Aventitious Angles.

Here's a list of 12 solutions. You can find solutions to other similar variants of the problem here .

Here's a solution and here's an animated solution. Here's one that uses trigonometry.

Here are some other sources: small pdf , website with some solutions , pdf with 8 solutions .

6 Helpful 0 Interesting 0 Brilliant 0 Confused

OMG! This problem is quite a famous one! Thanks for these sources.. :D

Satvik Golechha - 6 years, 10 months ago

Thank you, Sir Langley specialist.

Felipe Hofmann - 6 years, 10 months ago

I got 18 and 10 to start with. But it's confusing that more than one solution should exist, because to change x we have to dislocate the line EC a little, and so, if we try to produce a new solution of x going from an older one, within the same triangle, we are actually changing the angle ECB, which just doesn't make sense. Then it struck me that the only way it can be possible is if the size of the triangle is not specified, i.e the different solutions will be valid to different triangles with different 'sides' provided their internal angles are kept the same.

Pragjyotish Bhuyan Gogoi - 6 years, 10 months ago

we can use sine rule!! *I donno what the actual solution is!

Kartik Sharma - 6 years, 9 months ago

Pls can u tell me to do it without a calculator ?

Chirayu Bhardwaj - 5 years, 3 months ago

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